Maxmin equality with the concavified nonrevealing value
LLM status Partially solved unverified
C73 — Stochastic and Dynamic Games - Evolutionary Games - Repeated Games Economics catalogue AnthropicOpenAIopenaiclaude opus 5.5gpt 6 astra progpt 6 astra ultragpt-6 astragpt-6 astra pro 15% cataloguednot recordedPartially solvedStatement
GT-038 | Repeated games with incomplete information
P | Source-reported characterization question; a tail-measurable subclass is known
Definitions and mathematical statement
Let $K,I,J$ be finite. Nature draws $k\sim p\in\Delta(K)$ and reveals it only to the maximizing player; the players observe every subsequent joint action. For bounded Borel $f:K\times(I\times J)^{\mathbb N}\to\mathbb R$, let $\underline v_f(p)$ be the supremum over informed behavioral strategies $\sigma(k,h)$ of the infimum over uninformed strategies $\tau(h)$ of the expected payoff $f$.If the maximizing strategy also depends only on the public action history, it is nonrevealing. Define the complete-information functional \[ f_p(\omega)=\sum_{k\in K}p(k)f(k,\omega). \] The associated bounded-Borel Blackwell game has a value, denoted $u_f(p)$; it is the nonrevealing value. Define its concave envelope by \[ (\operatorname{cav}u_f)(p)= \sup\left\{\sum_{r=1}^{R}\lambda_r u_f(q_r): R\geq1,\ q_r\in\Delta(K),\ \lambda_r\geq0,\ \sum_r\lambda_r=1,\ \sum_r\lambda_rq_r=p\right\}. \] For fixed alphabets, characterize bounded Borel $f$ satisfying \[ \forall p\in\Delta(K),\qquad \underline v_f(p)=(\operatorname{cav}u_f)(p). \] The desired conditions should refer to the functional's structure and not simply repeat the equality to be characterized.
Short English statement
Exactly when is the informed player's guaranteed payoff the concavification of the nonrevealing value?Variants and scope boundaries
Nonrevealing strategies cannot condition directly on $k$. The equality concerns the maxmin, not necessarily the minimax or existence of a game value. Dependence on an initial finite block can matter for general Borel functionals. A one-prior characterization and an effective recognition algorithm would be different targets.Source relationship and status
[S1], Theorem 2.5, establishes this equality for its tail-measurable class. Section 5 asks for its characterization beyond that class. The known tail-measurable theorem is therefore a boundary of the open target, not itself an open conjecture.References
[S1] Gil Bar Castellon Koltun, Ehud Lehrer and Eilon Solan, The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs, arXiv:2512.01581 (2025). Locators: Definitions 2.1--2.4 and Theorem 2.5, pp. 6--7; Section 5, p. 18. https://arxiv.org/pdf/2512.01581
Common conventions: Source mathematical conventions
Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.
Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.
Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.
Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.
Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.
Statement licence: CC-BY-4.0 · Source file
Contributions 3
GPT 6 Astra Pro attempt on Econ-17
Research attempt and adversarial verification
9 October 2026 Original-target status: UNRESOLVED
C73-17: Maxmin equality with the concavified nonrevealing value
Input:C73-17.tex. Tools: web browsing [YES]; Python computation [YES]. 1) FORMAL RESTATEMENT
Let $K,I,J$ be nonempty finite sets, let $\mathbb N=\{0,1,\ldots\}$, and put $\Omega=(I\times J)^{\mathbb N}$ with its product Borel sigma-algebra. A payoff is a bounded Borel map $f:K\times\Omega\to\mathbb R$. The informed maximizer knows $k\sim p\in\Delta(K)$; both players observe all past joint actions, but choose the next actions simultaneously and independently conditional on their information. Write \[ V(p)=\sup_\sigma\inf_\tau\mathbb E_{p,\sigma,\tau}f, \qquad u(p)=\operatorname{val}\left(\sum_kp_kf(k,\cdot)\right). \] The second expression is the complete-information bounded-Borel Blackwell value. The external Blackwell determinacy theorem [martin-blackwell] states that, for two players with finite action sets, simultaneous moves, public observation of past joint actions, and a bounded Borel payoff of the infinite play, behavioral maxmin and minimax are equal. Each payoff $\sum_k p_kf(k,\cdot)$ meets these hypotheses. The theorem yields a value and $\eta$-optimal strategies for every $\eta>0$, not necessarily optimal-strategy attainment. Define \[ (\operatorname{cav}u)(p)=\sup_{p=\sum_{r=1}^R\lambda_rq_r} \sum_{r=1}^R\lambda_ru(q_r), \] where $R$ is a positive integer, $\lambda_r\geq0$, $\sum_r\lambda_r=1$, and $q_r\in\Delta(K)$.Literal target. Give a structural, noncircular characterization of \[ \mathcal C=\{f:\ \forall p\in\Delta(K),\ V(p)=(\operatorname{cav}u)(p)\}. \] This is not the assertion that every bounded Borel payoff belongs to $\mathcal C$. A payoff outside $\mathcal C$ therefore refutes that stronger assertion, not the characterization problem. No correction to the displayed target is needed. Priors on faces, singleton action sets, lack of optimal strategies, and distinctions between maxmin and minimax are stress points.
2) QUICK LITERATURE/CONTEXT CHECK
Castellon Koltun, Lehrer and Solan, Theorem 2.5 of [tail], prove equality for payoffs invariant under changes to finitely many action coordinates. Their Section 5 asks for a broader characterization. Their proof also records the universal lower bound proved below. This note does not claim novelty for that bound or reprove their tail-payoff theorem. All new calculations below concern finite-prefix examples and explicitly stated subclasses. The context check was performed on 9 October 2026 and is not an exhaustive exclusion of later work.3) ATTACK PLAN
Classification and tools. This is a repeated-game information problem. The useful tools are: finite convex decompositions (split a prior); behavioral realization (implement private mixtures without extra observable signals); equality of cylinder probabilities (identify infinite-play laws); concave envelopes (organize the lower bound); finite minimax (solve cylinder payoffs); posterior martingales (a possible route beyond cylinders); tail invariance (remove the cost of finite initial play).Proof tracks. First prove concavity of $V$ directly with behavioral strategies. Then identify structural subclasses where an uninformed upper bound matches it. Disproof tracks. Test a first-stage payoff, push the prior to the boundary, and make the informed best row depend on the type. The best tractable path is an exact two-type cylinder calculation plus the general lower bound.
4) WORK
In type zero row zero weakly dominates row one, and in type one row one weakly dominates row zero. If the informed player selects that type-dependent row, the payoffs against pure columns are $p$ and $1-p$, guaranteeing their minimum. Conversely, for an arbitrary informed strategy, column zero gives payoff at most $p$ and column one at most $1-p$. Selecting the better of these minimizing columns proves the matching upper bound. Later actions cannot affect a first-stage payoff. The functional is bounded and continuous on the product space; it is not tail invariant. All restrictions of the counterexample are thus verified. ∎
Small-case computation.
A singleton type has no information advantage. For the two-type example, exact rational evaluation at $p=0,1/4,1/2,3/4,1$ gives gaps $0,1/16,1/4,1/16,0$. The accompanying script evaluates:\begin{lstlisting} for p in [0, 1/4, 1/2, 3/4, 1]: # use exact fractions nonrevealing = p*(1-p) informed = min(p, 1-p) assert informed >= nonrevealing \end{lstlisting} These five checks supplement, rather than replace, the all-prior formula above.5) VERIFICATION
The mixture lemma was checked at zero-prior types and zero-likelihood histories; arbitrary prescribed actions there do not alter the laws. The proof uses $\eta$-optimal strategies rather than nonexistent maximizers. The cylinder example has exactly the monitoring and independent first-stage actions required by the input. Concavity concerns the maxmin, not equality with the minimax. The separable-type subclass is allowed to depend on an initial block, so tail invariance is not necessary for membership in $\mathcal C$. Conversely the continuous cylinder example shows that mere continuity or finite memory of the payoff does not suffice.\Needspace{8\baselineskip} 6) FINAL
LABEL: UNRESOLVED for the structural characterization in the attached file.Strongest checked partial results. A full proof of the universal lower bound; an exact bounded continuous two-type example with strict inequality at every interior prior; and two proved structural sufficient subclasses, one including non-tail functionals. The cylinder construction is a full counterexample to universal equality, not a solution of the actual classification target.
Exact first gap. No necessary-and-sufficient structural condition on a general bounded Borel $f$ has been established that ensures the reverse inequality $V(p)\leq\operatorname{cav}u(p)$ for every $p$.
Top three next moves. (1) Classify finite-prefix matrix trees by recursively tracking the posterior and nonrevealing continuation game. (2) Determine a quantitative condition under which altering an arbitrary finite prefix has uniformly negligible payoff effect. (3) Test mixtures of a tail-invariant functional and the displayed first-stage matrix for a sharp boundary between equality and a positive information gap.
Minimal counterexample perspective. Two types, two actions each and one payoff-relevant stage already disprove the broadest candidate sufficient condition (bounded continuity). They do not falsify an as-yet-unspecified structural characterization.
Reproducibility and scope.
This is one of eleven companion reports. The bundle includesverify.py and its executed results. Finite experiments supplement the proofs; they are not proofs of asymptotic claims. Literature coverage is targeted, and no novelty or formal proof-assistant certification is claimed.References
- [martin-blackwell] Donald A. Martin. The Determinacy of Blackwell Games. The Journal of Symbolic Logic 63(4) (1998), 1565--1581. https://www1.cmc.edu/pages/faculty/MONeill/math188/papers/martin2.pdf.
- [tail] Gil Bar Castellon Koltun, Ehud Lehrer, and Eilon Solan. The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs. arXiv:2512.01581 (2025), Theorem 2.5 and Sections 3 and 5. https://arxiv.org/html/2512.01581.
GPT 6 Astra Ultra attempt on Econ-17
Maxmin equality with the concavified nonrevealing value GPT 6 Astra Ultra
Target and contribution
The target asks for structural conditions on a bounded Borel payoff functional $f(k,\omega)$ ensuring $\underline v_f(p)=({\operatorname{cav}} u_f)(p)$ for every prior. Here $\underline v_f$ is the informed maxmin and $u_f$ the nonrevealing value. The known tail-measurable theorem is already outside the open portion of the question [tail]. This attempt proves the general direction of the inequality carefully, then derives a complete criterion for a small first-stage family and a sufficient condition for general finite first-stage games. These are elementary restricted results, not a new general characterization.Why a counterexample can only lie above concavification
To justify this construction within the stated behavioral-strategy convention, rather than implicitly granting an extra remembered signal, define at public history $h=(i_1,j_1,\ldots,i_t,j_t)$ the weights \[ w_r(k,h)=\frac{\lambda_rq_r(k)}{p(k)} \prod_{s=1}^t\sigma^r(i_s\mid k,h_{s-1}). \] At a history with positive total weight, play the normalized weighted average of the next-action distributions $\sigma^r(\cdot\mid k,h)$. At a zero-weight history choose arbitrarily. Multiplying these conditional probabilities along a finite history telescopes the normalizing weights and produces exactly the mixture of the $\sigma^r$ likelihoods. The likelihood contributions of the opponent are common to every $r$ and cancel in the posterior weights. Hence this behavioral strategy induces the same probabilities on every finite cylinder as the private-index construction, against every opponent. The laws on the infinite product coincide, so bounded Borel payoffs also have equal expectations. States with $p(k)=0$ are irrelevant and may use arbitrary prescriptions. Letting $\epsilon\downarrow0$ proves concavity.
The informed player may ignore its state. For any state-independent strategy pair the expected payoff is the payoff in the complete-information functional $f_p$. Using an $\epsilon$-optimal maximizing strategy for that game gives $\underline v_f(p)\ge u_f(p)$. Apply the concavity just proved to every finite splitting of $p$, replace each $\underline v_f(q_r)$ by $u_f(q_r)$, and take the supremum defining ${\operatorname{cav}} u_f$. ∎
The inequality is known in the source; the proof is included to make the quantifiers and the admissibility of hidden randomization explicit. No claimed minimax equality for the incomplete-information Borel game is needed.
A fully solved first-stage parameter family
Fix states $K=\{0,1\}$ and two actions for each player. Let $p$ be the probability of state $0$. For parameters $a,b\ge0$, let the payoff depend only on the first joint action and be given by the matrices \[ A_0=\begin{pmatrix}a&0\\0&0\end{pmatrix},\qquad A_1=\begin{pmatrix}0&0\\0&b\end{pmatrix}. \] Subsequent actions have no payoff effect, so these are bounded Borel functionals, generally not tail measurable.For $a,b>0$, differentiation gives \[ u_f''(p)=-\frac{2a^2b^2}{[b+(a-b)p]^3}<0. \] Thus $u_f$ is concave on the closed interval, by its continuous endpoint extension, and its concavification is itself.
The informed player can choose row one in state $0$ and row two in state $1$. Against a column-one probability $y$ its expected payoff is $Ay+B(1-y)$, whose minimum over $y\in[0,1]$ is $\min\{A,B\}$. Conversely the column player may choose column one, guaranteeing payoff at most $A$ against every informed row strategy, or column two, guaranteeing at most $B$. This proves the informed value. If $A,B>0$, then $AB/(A+B)$ is strictly below both $A$ and $B$. At every interior prior they are both positive, proving the strict inequality. If $a=0$ or $b=0$, a column gives zero in both states against all rows; all payoffs are nonnegative, so all three quantities are identically zero. These cases prove the claimed if-and-only-if condition. ∎
For $a=b=1$ and $p=1/2$, the exact comparison is $\underline v_f=1/2$ and ${\operatorname{cav}} u_f=1/4$. This two-state, two-action calculation is a small adversarial check on any proposed extension from tail payoffs to arbitrary finite-prefix dependence.
A sufficient condition allowing prefix dependence
This condition is an explicit finite matrix certificate. It includes non-tail payoffs, so tail measurability is not necessary. It is not asserted necessary: the general characterization must account for other mechanisms as well.
Remaining gap
The general lower bound, the exact first-stage parameter classification, and the common-minimizer certificate do not provide necessary and sufficient structural conditions for arbitrary bounded Borel functionals. In particular, concatenating a finite information-dependent block with a tail game may change both the available signals and the payoff. No invariance under such concatenation is assumed or proved.References
- [tail] G. Bar Castellon Koltun, E. Lehrer, and E. Solan, The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs, arXiv:2512.01581 (2025), Definitions 2.1--2.4, Theorem 2.5, the first part of its proof, and Section 5. https://arxiv.org/pdf/2512.01581. The primary PDF, its tail-measurable equality, and its general lower-bound discussion were checked on 9 October 2026. The results in this attempt are proved directly; no existence of an incomplete-information game value is inferred from the tail theorem.
Completion Estimate
15%C73-17: An exact local-geometry characterization of maxmin concavification
C73-17: An exact local-geometry characterization
of maxmin concavification
Ji Ho Bae
LawAll AI
10 October 2026
The criterion is an infinitary local-capital certificate. Conserved likelihood flows encode the informed player's likelihoods; explicit behavioral action kernels, expected-payoff optimization, and game values are absent from its definition. Blackwell Borel determinacy is used only for nonrevealing games, and Borel superhedging duality for all-path capital bounds. No value of the incomplete-information game is asserted.
Target and notation
Fix nonempty finite sets $K,I,J$, put $m=|K|$, $X=I\times J$, $H=\bigcup_{t\ge0}X^t$, and $\Omega=X^{\mathbb N}$. The empty history is $\varnothing$, $hij$ means append the pair $(i,j)$, and $\omega|t$ is the length-$t$ prefix. Let $f:K\times\Omega\to\mathbb R$ be bounded and Borel. The informed maximizer has behavioral prescriptions $\sigma_k(i\mid h)$; the uninformed minimizer has prescriptions $\tau_j(h)$, with simultaneous independent current randomizations and full observation of past joint actions.
Write $L_f(p)=\sup_\sigma\inf_\tau\mathbb E_{p,\sigma,\tau} f$. The nonrevealing payoff is $f_q(\omega)=\sum_kq_k f(k,\omega)$, whose bounded-Borel Blackwell game has a value $u_f(q)$. Define $c_f=\operatorname{cav}u_f$ by the finite-splitting formula in the canonical statement. The target is to characterize the payoffs satisfying $L_f(p)=c_f(p)$ for every $p\in\Delta(K)$.
Choose $a<b$ with $a\le f\le b$, and put $F=(f-a)/(b-a)$, so $0\le F\le1$. Write $F_k(\omega)=F(k,\omega)$ and $F_q=\sum_kq_kF_k$. The target is unchanged by this normalization. Let $\Delta_{\mathbb Q}(K)$ denote priors with rational coordinates. All vectors and arrays below are real-valued; no rationality restriction is imposed on witnesses.
Indeed every strategy pair's payoff transforms as $a+(b-a)\mathbb E F$; positive affine transformations commute with the prescribed supremum and infimum, so $L_f=a+(b-a)L_F$ and $u_f=a+(b-a)u_F$. Every splitting has weights summing to one, which also gives $c_f=a+(b-a)c_F$. This proves the normalization assertion, even for constant $f$, for which any enclosing bounds $a<b$ may be used.
\clearpage
The local-geometry certificate condition
A likelihood flow at $p$ is an array $z_k(h)\ge0$ satisfying
\begin{equation} z_k(\varnothing)=p_k,\qquad z_k(hij)\text{ is independent of }j,\qquad \sum_{i\in I}z_k(hij)=z_k(h) \label{wsref-0-65-71-3a-66-6c-6f-77} \end{equation}
for every $k,h,j$. In particular every entry is at most $p_k$; if $z_k(h)=0$, all its children are zero. These are direct algebraic identities, including on histories assigned zero flow.
For a prior $p$ and integer $n\ge1$, the condition requires the following quantifier order.
First choose one lower-capital package consisting of an integer $1\le R\le m$, positive weights $\lambda_r$, vectors $q_r\in\Delta(K)$, and arrays $C_r:H\to[0,1]$, such that
\begin{equation} \sum_r\lambda_r=1,\qquad \sum_r\lambda_rq_r=p. \label{wsref-0-65-71-3a-73-70-6c-69-74} \end{equation}
At each $h$ and for each $r$, form row increment vectors
\[ D^r_i(h)=\big(C_r(hij)-C_r(h)\big)_{j\in J}\in\mathbb R^J. \]
Require the finite convex-hull intersection
\begin{equation} \operatorname{conv}\{D^r_i(h):i\in I\} \cap\mathbb R_-^J\ne\varnothing \label{wsref-0-65-71-3a-6c-6f-77-65-72-5f-68-75-6c-6c} \end{equation}
and the boundary inequality on every infinite path
\begin{equation} \liminf_{t\to\infty}C_r(\omega|t)\ge1-F_{q_r}(\omega). \label{wsref-0-65-71-3a-6c-6f-77-65-72-5f-62-6f-75-6e-64-61-72-79} \end{equation}
Its root reference is $\beta=\sum_r\lambda_r(1-C_r(\varnothing))$.
Then, for every likelihood flow $z$ satisfying (eq:flow), choose upper-capital arrays $A_k:H\to[0,1]$. At each $h$, form column increment vectors
\[ V^z_j(h)=\left( \sum_{i\in I}z_k(hij)\,[A_k(hij)-A_k(h)] \right)_{k\in K}\in\mathbb R^K. \]
Require
\begin{equation} \operatorname{conv}\{V^z_j(h):j\in J\} \cap\mathbb R_-^K\ne\varnothing \label{wsref-0-65-71-3a-75-70-70-65-72-5f-68-75-6c-6c} \end{equation}
and, again on every path and for every state,
\begin{equation} \liminf_{t\to\infty}A_k(\omega|t)\ge F_k(\omega). \label{wsref-0-65-71-3a-75-70-70-65-72-5f-62-6f-75-6e-64-61-72-79} \end{equation}
Finally require the root inequality
\begin{equation} \sum_kp_kA_k(\varnothing)\le\beta+1/n. \label{wsref-0-65-71-3a-72-6f-6f-74-5f-62-75-64-67-65-74} \end{equation}
The payoff condition is:
(G) For each $p\in\Delta_{\mathbb Q}(K)$ and $n\ge1$, choose one package satisfying (eq:split)--(eq:lower_boundary). For every flow $z$ satisfying (eq:flow), there must then exist capitals $A_k$ satisfying (eq:upper_hull)--(eq:root_budget).
The lower package is fixed before $z$, whereas the upper arrays may depend on $z$. Likelihood flows preserve the informed-strategy quantification. The criterion uses no explicit behavioral kernels, expected-payoff optimization, or game values: its inputs are path values of $F$, and its tests are conserved flows, finite convex geometry, and pointwise boundary and root inequalities.
\[ \forall p\in\Delta(K),\quad L_f(p)=(\operatorname{cav}u_f)(p) \quad\Longleftrightarrow\quad \text{(G)}. \]
The result covers singleton alphabets, simplex faces, arbitrary finite-prefix dependence, and absence of optimal-strategy attainment. The choice of normalization bounds does not affect the equivalence.
Established results and the bounded-capital reduction
Blackwell Borel determinacy [Martin, Theorem 10, p. 1578] supplies existence of the nonrevealing value: the game has finite action sets, simultaneous moves, complete observation of past joint actions, and bounded Borel payoff $F_q$. We use it only for these complete-information games. We do not invoke determinacy for the incomplete-information game.
The second established input is a finite-tree Borel superhedging theorem. Assign each $h\in H$ a nonempty closed convex set $P_h\subseteq\Delta(X)$. For any Borel $W:\Omega\to[0,1]$, let
\[ c=\sup_{\pi_h\in P_h\ \forall h}\mathbb E_\pi W, \]
where the expectation uses the precise probability tree selected by all its local kernels. Then for each $\delta>0$ there is a globally bounded-below, real-valued array $M:H\to\mathbb R$ satisfying
\[ \sup_{\pi\in P_h}\sum_{x\in X}\pi_x M(hx)\le M(h),\qquad \liminf_tM(\omega|t)\ge W(\omega)\quad(\omega\in\Omega), \]
and $M(\varnothing)<c+\delta$. This is the combination of definition $3$ on p. 314, Theorem 2 on p. 315, and Theorem 13 on p. 318 in T'Joens and De Bock [TDB]. Theorem 2 identifies the axiomatic upper expectation with the game-theoretic superhedging expectation; Theorem 13 identifies the latter with the Borel measure upper expectation. The product sigma field is the Borel sigma field because the alphabet is finite. The theorem has no regularity assumption on the nodewise credal sets beyond the stated finite-dimensional hypotheses.
The capital may always be taken in $[0,1]$. First $M(h)\ge0$ at every $h$: otherwise the local inequality and nonemptiness of $P_h$ give a child with $M(hx)\le M(h)<0$. Repeating this produces an infinite path whose liminf is negative, contrary to the terminal inequality. Next put $M'=\min(M,1)$. For every $\pi\in P_h$, concavity and monotonicity of $s\mapsto\min(s,1)$ give
\[ \sum_x\pi_x\min(M(hx),1) \le\min\!\left(\sum_x\pi_xM(hx),1\right) \le\min(M(h),1). \]
Its liminf is $\min(\liminf M,1)$, which still dominates $W$, and its root cost can only decrease. This proves the claimed bounded-capital version, including terminal domination on paths of probability zero.
Flows and behavioral likelihoods
Given any informed behavioral strategy $\sigma$, define
\begin{equation} z_k(h)=p_k\prod_{s<t}\sigma_k(i_s\mid h_s), \qquad h=((i_0,j_0),\ldots,(i_{t-1},j_{t-1})). \label{wsref-0-65-71-3a-66-6c-6f-77-5f-6c-69-6b-65-6c-69-68-6f-6f-64} \end{equation}
These entries satisfy (eq:flow). The opponent's likelihood is deliberately absent.
Conversely, given a flow $z$, choose a fixed distribution $x^0\in\Delta(I)$ and set
\begin{equation} \sigma_k(i\mid h)= \begin{cases} z_k(hij)/z_k(h),&z_k(h)>0,\\ x^0_i,&z_k(h)=0. \end{cases} \label{wsref-0-65-71-3a-66-6c-6f-77-5f-72-65-63-6f-6e-73-74-72-75-63-74-69-6f-6e} \end{equation}
The quotient is independent of $j$, and the conservation identity makes it a probability vector. Induction proves (eq:flow_likelihood) at every history. In the zero case all children are zero, so the induction also holds there. At zero-prior states it holds because all entries are zero.
Against any uninformed strategy $\tau$, the probability of the cylinder fixing state $k$ and public history $h$ is exactly
\begin{equation} z_k(h)\prod_{s<t}\tau_{j_s}(h_s). \label{wsref-0-65-71-3a-63-79-6c-69-6e-64-65-72-5f-6c-61-77} \end{equation}
Thus flows encode the exact statewise likelihood structure with the opponent's current information restriction. The identities are valid off the support as well as on it.
If $z$ was encoded from a strategy, reconstruction can change that strategy only at state-history nodes with $z_k(h)=0$. Formula (eq:cylinder_law) then gives identical state-public cylinder probabilities for the original and reconstructed strategies against every opponent. They therefore induce identical Borel laws and payoff expectations. This also justifies using the reconstruction in upper bounds for an arbitrary originally supplied strategy.
Soundness of each lower package
For fixed $r,h$, (eq:lower_hull) supplies coefficients $x_i(h)\ge0$, summing to one, with
\begin{equation} \sum_i x_i(h)C_r(hij)\le C_r(h)\qquad(j\in J). \label{wsref-0-65-71-3a-6c-6f-77-65-72-5f-73-74-65-70} \end{equation}
Choose such coefficients at every history. Since $H$ is countable, this defines a behavioral nonrevealing strategy without any measurable-selection issue. Against every opponent $\tau$, the process $C_r(\omega|t)$ is a bounded nonnegative supermartingale: (eq:lower_step) holds for each current pure column, and averaging over $\tau(h)$ preserves it. By (eq:lower_boundary) and Fatou's lemma,
\[ \mathbb E_{x,\tau}(1-F_{q_r}) \le\mathbb E_{x,\tau}\liminf_t C_r(\omega|t) \le\liminf_t\mathbb E_{x,\tau}C_r(\omega|t) \le C_r(\varnothing). \]
Therefore $u_F(q_r)\ge1-C_r(\varnothing)$. Weighting the inequalities and using (eq:split) gives
\begin{equation} \beta\le\sum_r\lambda_r u_F(q_r)\le c_F(p). \label{wsref-0-65-71-3a-72-65-66-65-72-65-6e-63-65-5f-62-6f-75-6e-64} \end{equation}
This bound uses only the package already fixed; it is independent of every subsequently supplied flow.
In fact, for every bounded Borel $F$ and every prior $p$, the supremum of the root references $\beta$ over all packages (eq:split), (eq:lower_hull), and (eq:lower_boundary) with $R\le m$ equals $c_F(p)$. The displayed bound proves one direction. Section 9 gives an exact maximizing split, and the lower-capital construction in the first half of Section 10 uses neither $L_F=c_F$ nor any particular flow; applied to that split, it gives $\beta>c_F(p)-2\delta$ for every $\delta>0$. This proves the reverse direction. Thus the lower package is itself a value-free geometric representation of concavification.
Soundness of the upper-capital response
Fix a flow $z$, its reconstructed strategy $\sigma$, and arrays $A_k$ satisfying (eq:upper_hull) and (eq:upper_boundary). The hull condition supplies a single column distribution $y(h)\in\Delta(J)$ at every $h$, with
\begin{equation} \sum_jy_j(h)\sum_i z_k(hij) [A_k(hij)-A_k(h)]\le0\qquad(k\in K). \label{wsref-0-65-71-3a-72-65-73-70-6f-6e-73-65-5f-77-65-69-67-68-74-65-64-5f-73-74-65-70} \end{equation}
For $z_k(h)>0$, division by $z_k(h)$, followed by (eq:flow_reconstruction), gives
\begin{equation} \sum_{i,j}\sigma_k(i\mid h)y_j(h)A_k(hij)\le A_k(h). \label{wsref-0-65-71-3a-72-65-73-70-6f-6e-73-65-5f-73-74-65-70} \end{equation}
If $p_k>0$, every positive-probability history under the conditional law in state $k$ has $z_k(h)>0$, by (eq:cylinder_law). Consequently $A_k(\omega|t)$ is a bounded nonnegative supermartingale under that conditional law. No condition after division is needed at a state-history cylinder of probability zero. The boundary (eq:upper_boundary), which holds everywhere, and Fatou yield
\[ \mathbb E_{k,\sigma_k,y}F_k\le A_k(\varnothing). \]
Multiply by $p_k$ and sum over its positive coordinates; zero-prior states contribute nothing. This proves
\begin{equation} \inf_\tau\mathbb E_{p,\sigma,\tau}F \le\mathbb E_{p,\sigma,y}F \le\sum_kp_k A_k(\varnothing). \label{wsref-0-65-71-3a-72-65-73-70-6f-6e-73-65-5f-62-6f-75-6e-64} \end{equation}
The same $y(h)$ works for all states at that node. Its current randomization cannot depend on the hidden state or the current informed action. This information restriction is precisely the common convex combination in (eq:upper_hull).
A behavioral proof of the universal lower bound
For every bounded Borel payoff, $L_F\ge c_F$. Here is a proof that does not rely on tail measurability or public revelation of a split.
Take any finite splitting $p=\sum_r\lambda_rq_r$, any informed strategies $\sigma^r$, and a state $k$ with $p_k>0$. Put $\alpha_r(k)=\lambda_rq_{r,k}/p_k$,
\[ L_r(k,h)=\prod_{s<t}\sigma^r_k(i_s\mid h_s),\qquad D(k,h)=\sum_r\alpha_r(k)L_r(k,h). \]
At positive $D(k,h)$ prescribe
\[ \sigma_k(i\mid h)= \frac{\sum_r\alpha_r(k)L_r(k,h)\sigma^r_k(i\mid h)}{D(k,h)}; \]
at zero $D$, or zero $p_k$, prescribe any fixed distribution. The numerator is the next child $D(k,hi j)$, independent of the current $j$. Multiplication telescopes, giving $\prod_{s<t}\sigma_k(i_s\mid h_s)=D(k,h)$ at every history, including zero likelihoods. For any common opponent $\tau$, its factor $\prod_{s<t}\tau_{j_s}(h_s)$ is independent of $r$. Hence the constructed law satisfies, on every state-public cylinder and thus on the generated Borel sigma field,
\begin{equation} \mathbb P_{p,\sigma,\tau} =\sum_r\lambda_r\mathbb P_{q_r,\sigma^r,\tau}. \label{wsref-0-65-71-3a-6d-69-78-74-75-72-65-5f-6c-61-77} \end{equation}
For each $r$, choose $\sigma^r$ guaranteeing $L_F(q_r)-\varepsilon$. Such approximate strategies exist by the definition of supremum; no attainment is assumed. Equation (eq:mixture_law) implies $L_F(p)\ge\sum_r\lambda_rL_F(q_r)-\varepsilon$. Letting $\varepsilon\downarrow0$ proves concavity of $L_F$. Ignoring $k$ gives $L_F(q)\ge u_F(q)$ for each $q$. Therefore $L_F(p)\ge\sum_r\lambda_ru_F(q_r)$ for every split, and taking their supremum gives
\begin{equation} L_F(p)\ge c_F(p). \label{wsref-0-65-71-3a-75-6e-69-76-65-72-73-61-6c-5f-6c-6f-77-65-72} \end{equation}
This is the universal lower bound recorded in Part 1 of the proof of the source theorem [KLS].
Sufficiency at rational priors
Assume (G), and fix rational $p$ and integer $n\ge1$. Take the common package. Every informed $\sigma$ has a flow $z$, so (G) supplies upper arrays for it. Equations (eq:response_bound), (eq:root_budget), and (eq:reference_bound) give
\[ \inf_\tau\mathbb E_{p,\sigma,\tau}F \le\sum_kp_kA_k(\varnothing) \le\beta+1/n\le c_F(p)+1/n. \]
Taking the supremum over $\sigma$, and then letting $n\to\infty$, proves $L_F(p)\le c_F(p)$. Combine with (eq:universal_lower) to obtain equality at every rational prior.
A maximizing splitting exists with at most $|K|$ terms
For every finite splitting and numbers $a_r\in\mathbb R$, another splitting using a subset of the same $q_r$ has at most $m$ positive weights and has weighted $a_r$-sum at least as large. Remove zero weights. If more than $m$ remain, the $q_r\in\mathbb R^m$ are linearly dependent. Choose nonzero $v=(v_r)$ with $\sum_r v_rq_r=0$. Because every $q_r$ sums to one, $\sum_r v_r=0$; therefore $v$ has both signs. Reverse its sign if necessary so $\sum_rv_ra_r\ge0$, and set
\[ t=\min_{v_r<0}\frac{\lambda_r}{-v_r}>0,\qquad \lambda'_r=\lambda_r+tv_r. \]
The new weights are nonnegative, conserve the barycenter and their sum, make at least one formerly positive weight zero, and do not reduce the objective. Iterating finitely many times leaves at most $m$ weights. Apply this with $a_r=u_F(q_r)$. Thus the supremum defining $c_F(p)$ is unchanged if restricted to at most $m$ terms.
The function $u_F$ is 1-Lipschitz and in particular continuous; Section 11 proves this directly without using splitting attainment. Pad any splitting of at most $m$ terms with zero weights to exactly $m$ terms. The feasible set
\[ \{(\lambda,q_1,\ldots,q_m)\in\Delta(\{1,\ldots,m\})\times\Delta(K)^m: \sum_{r=1}^m\lambda_rq_r=p\} \]
is nonempty and compact, since the barycenter equality is closed. The continuous objective $\sum_r\lambda_ru_F(q_r)$ attains a maximum there. By support reduction its maximum equals $c_F(p)$. Discarding the zero weights gives an exact maximizing split with $1\le R\le m$ positive terms. This finite-dimensional attainment does not assert that any infinite-game strategy attains its optimization.
Necessity: one common lower package, then all flows
Assume $L_F(p)=c_F(p)$ for every prior, and fix a prior $p$ and integer $n\ge1$. Write $c=c_F(p)=L_F(p)$ and choose $\delta=1/(5n)$.
Choose an exact maximizing split with $R\le m$ and $\sum_r\lambda_ru_F(q_r)=c$, using Section 9. For each $r$, choose a nonrevealing strategy $x^r(h)\in\Delta(I)$ such that
\begin{equation} \inf_\tau\mathbb E_{x^r,\tau}F_{q_r}\ge u_F(q_r)-\delta. \label{wsref-0-65-71-3a-6e-6f-6e-72-65-76-65-61-6c-69-6e-67-5f-67-75-61-72-61-6e-74-65-65} \end{equation}
At node $h$, use the credal set
\[ P^r_h=\{(x^r_i(h)b_j)_{ij}:b\in\Delta(J)\}. \]
It is nonempty, closed, and convex, as the continuous linear image of a finite simplex. A compatible precise tree corresponds exactly to an uninformed opponent against $x^r$: its $J$-marginal recovers $b$, and its kernel is the displayed independent product. This statement holds on all nodes; assignments on root-null nodes do not affect the root law. For $W=1-F_{q_r}$, the robust expectation is
\[ \sup_\tau\mathbb E_{x^r,\tau}(1-F_{q_r}) =1-\inf_\tau\mathbb E_{x^r,\tau}F_{q_r} \le1-u_F(q_r)+\delta. \]
The bounded-capital duality of Section 3 supplies $C_r\in[0,1]$ with (eq:lower_boundary),
\[ \sum_i x^r_i(h)C_r(hij)\le C_r(h)\quad(j\in J), \qquad C_r(\varnothing)<1-u_F(q_r)+2\delta. \]
The coefficients $x^r(h)$ verify (eq:lower_hull). This package depends only on $p,n$, and its root reference satisfies
\begin{equation} \beta=\sum_r\lambda_r(1-C_r(\varnothing)) >\sum_r\lambda_r u_F(q_r)-2\delta=c-2\delta. \label{wsref-0-65-71-3a-72-65-66-65-72-65-6e-63-65-5f-63-6f-73-74} \end{equation}
Now let $z$ be any likelihood flow. Reconstruct its informed strategy $\sigma$ by (eq:flow_reconstruction). Since the infimum against this particular $\sigma$ is at most the maxmin $L_F(p)=c$, choose a response $\tau$ with
\begin{equation} \gamma:=\mathbb E_{p,\sigma,\tau}F<c+\delta. \label{wsref-0-65-71-3a-74-61-69-6c-6f-72-65-64-5f-72-65-73-70-6f-6e-73-65} \end{equation}
For each state $k$, apply the duality to the singleton precise credal sets
\[ P^{k}_h=\{(\sigma_k(i\mid h)\tau_j(h))_{ij}\}. \]
Singletons are nonempty, closed, and convex. Their unique compatible precise tree is the fixed-state-$k$ kernel law against $\sigma_k,\tau$, defined independently of whether $p_k>0$. For $W=F_k$, obtain $A_k\in[0,1]$ satisfying (eq:upper_boundary),
\begin{equation} \sum_{i,j}\sigma_k(i\mid h)\tau_j(h)A_k(hij)\le A_k(h), \qquad A_k(\varnothing)<\mathbb E_{k,\sigma_k,\tau}F_k+\delta. \label{wsref-0-65-71-3a-70-72-65-63-69-73-65-5f-68-65-64-67-65} \end{equation}
Multiplying the local inequality by $z_k(h)$, and using $z_k(hij)=z_k(h)\sigma_k(i\mid h)$ everywhere, shows that the single common coefficients $\tau_j(h)$ verify (eq:upper_hull). When $z_k(h)=0$, both sides of this multiplied inequality are zero; no division has occurred. Weighting the root estimates gives
\[ \sum_kp_kA_k(\varnothing)<\gamma+\delta<c+2\delta <\beta+4\delta<\beta+1/n. \]
Thus (eq:root_budget) holds for this flow. The package was already fixed and the argument applies to every flow. Necessity holds at every prior, in particular at all rational priors required in (G).
Extension from rational priors to every prior
Both $L_F$ and $u_F$ are 1-Lipschitz for the $\ell^1$ distance. For a fixed admissible strategy pair, its expected payoff at $p$ has the form $\sum_kp_ka_k$, where $a_k\in[0,1]$ is the conditional expectation under that pair. It follows that its change is at most $\|p-p'\|_1$. The strategy spaces are independent of the prior, including prescriptions at zero-prior states. Taking either relevant supremum/infimum preserves the same uniform Lipschitz bound.
Here is a proof that $c_F=\operatorname{cav}u_F$ is also 1-Lipschitz, including the boundary. Given $p,p'$, choose a coupling matrix $T_{ab}\ge0$ with row sums $p_a$, column sums $p'_b$, and diagonal entries $T_{aa}=\min(p_a,p'_a)$. Explicitly, put these entries on the diagonal; the nonnegative residual row and column masses have disjoint supports and equal total $d=\tfrac12\|p-p'\|_1$. If $d>0$, couple them by their product divided by $d$; if $d=0$, no residual is needed. For $p_a>0$, let $S_{ab}=T_{ab}/p_a$, and choose any probability row when $p_a=0$.
Transport any finite splitting $p=\sum_r\lambda_rq_r$ by setting $q'_r=q_r S$. Then $\sum_r\lambda_rq'_r=p'$. Moreover,
\[ \begin{aligned} \sum_r\lambda_r\|q'_r-q_r\|_1 &\le\sum_r\lambda_r\sum_aq_{r,a}\|S_{a\cdot}-e_a\|_1\\ &=\sum_ap_a\,2(1-S_{aa}) =2\sum_a(p_a-T_{aa})=\|p-p'\|_1. \end{aligned} \]
Terms with $p_a=0$ vanish: for every positive $\lambda_r$, barycenter conservation implies $q_{r,a}=0$. By the Lipschitz property of $u_F$,
\[ \sum_r\lambda_ru_F(q'_r) \ge\sum_r\lambda_ru_F(q_r)-\|p-p'\|_1. \]
Take the supremum over all splits on the right. This gives $c_F(p')\ge c_F(p)-\|p-p'\|_1$. Interchanging $p,p'$ proves the opposite bound. Rational priors are dense in the whole simplex. Equality of these two continuous functions at rational priors therefore extends to every prior, completing sufficiency and the theorem.
Scope
All quantified functionals are bounded Borel; no tail, continuity, prefix, or finite-horizon restriction is added. The terminal constraints hold on every path, including paths assigned probability zero by every relevant root law. The Borel duality theorem is used to prove their necessity. No optimizing game strategy needs to exist; the finite-dimensional prior splitting does attain its maximum, and this is proved in Section 9.
The condition tests the functional against real arrays by flow conservation, finite convex geometry, and pointwise boundary comparisons. The common column hull encodes the uninformed player's information constraint. The likelihood flows are equivalent to the informed player's likelihoods, as proved algebraically. The extra capital arrays are dual certificates with local inequalities and global boundary domination.
The characterization is infinitary. An arbitrary Borel functional need not have a finite input encoding, and its all-path boundary comparisons are not asserted decidable or finitely checkable. The theorem does not claim a simpler qualitative classification or an effective algorithm. Finite recognition of an arbitrary Borel functional and simpler qualitative subclass criteria are separate questions.
The condition concerns the maxmin alone. Since the upper arrays may depend on the tested flow, the response distribution extracted from (eq:upper_hull) may also depend on that flow. There is no common response against all informed strategies. In particular the proof makes no inference that the uninformed minimax equals the maxmin. Known tail-payoff examples where the maxmin equals concavification but a game value fails to exist are compatible with this quantifier order.
Boundary and first-stage examples
If $F_k(\omega)=a_k\in[0,1]$ is constant in the public path, choose one split with $q_1=p$, $\lambda_1=1$, constant $C_1=1-\sum_kp_ka_k$, and, for every flow, constant $A_k=a_k$. All row and column increment vectors vanish. Both boundary conditions hold with equality, and the root inequality holds with zero error. Thus the full certificate includes all statewise constant payoffs, including simplex-face priors and singleton action sets.
The condition does not accept every bounded continuous payoff. For $K=I=J=\{0,1\}$, let $F$ depend only on the first joint action and have state matrices
\[ A^0=\begin{pmatrix}1&0\\0&0\end{pmatrix}, \qquad A^1=\begin{pmatrix}0&0\\0&1\end{pmatrix}. \]
Writing $p=\Pr(k=0)$, the nonrevealing matrix is $\operatorname{diag}(p,1-p)$. A row-zero probability $x$ gives column payoffs $px$ and $(1-p)(1-x)$. Their minimum is maximized at $x=1-p$ for $0<p<1$, giving $u_F(p)=p(1-p)$. The column distribution that chooses column zero with probability $1-p$ gives both rows this same payoff, proving the matching upper bound. At endpoints the value is zero. The function $p(1-p)$ is concave, so its concavification is itself.
The informed choice of row zero in state zero and row one in state one gives payoffs $p$ and $1-p$ against the two columns. Conversely pure column zero bounds every informed strategy by $p$, and pure column one bounds it by $1-p$. Therefore $L_F(p)=\min(p,1-p)$. At $p=1/2$, the type-dependent first action gives expected payoff $1/2$ against every opponent. Its associated flow, tested in (G), forces every upper root cost to be at least $1/2$, by Section 6. Every lower-package reference is at most $1/4$, by Section 5. Hence (eq:root_budget) is impossible for $n\ge5$.
Use of AI tools
The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.References
- [KLS] Gil Bar Castellon Koltun, Ehud Lehrer, and Eilon Solan. The Maxmin Value of Repeated Games with Incomplete Information on One Side and Tail-Measurable Payoffs. arXiv:2512.01581 $2025$. Definition 2.1, p. 6, specifies the game; Remark 2.2 and Definitions 2.3--2.4, p. 7, specify the evaluation and nonrevealing game. Theorem 2.5, p. 8, states the tail-payoff theorem; Section 3, Part 1 of its proof, p. 8, records the universal lower bound. Section 5, p. 18, asks the characterization question. https://arxiv.org/pdf/2512.01581v1.
- [Martin] Donald A. Martin. The Determinacy of Blackwell Games. Journal of Symbolic Logic 63$4$, 1565--1581 $1998$. The behavioral finite-action setup is given on p. 1567. Theorem 10, p. 1578, supplies the bounded-Borel Blackwell determinacy. https://doi.org/10.2307/2586667. https://www1.cmc.edu/pages/faculty/MONeill/math188/papers/martin2.pdf.
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